My favourite and least favourite math teacher

 The best math teacher I had was my high school math teacher. He explained everything in great detail. Before introducing a new topic, he would review all the prerequisite knowledge that we needed and make sure that we understood it before moving on. During his explanations, he often proved important results instead of simply asking us to accept them, and sometimes he would even show us more than one way to prove the same result. After teaching a new concept, he would also explain how it was connected to other parts of mathematics and where it could be applied. Looking back, I realize that my own teaching style has been strongly influenced by him. I also like to review the necessary background first, explain why something works, and help students see connections between different ideas instead of teaching each topic separately.

The least effective math teaching I have encountered was actually not from one of my own teachers. One of my former students moved back to China because he was having difficulty in English class. Instead of attending a regular high school, he studied A-level courses at a tutoring school. He recently complained to me that his math teacher's explanations were very confusing. Since I encouraged him to respect his teacher, work hard, and figure out a way to improve himself, he sent me a short audio recording of a statistics lesson. To be honest, after listening to the recording, although I still try to encourage my student without saying anything negative about his teacher, I could understand why he was struggling. The teacher was explaining a statistics problem, but he rarely repeated the names of important concepts such as Q1, median and Q3, and it did not sound like he was writing the important expressions clearly on the board (given the short time he left for the students to think). This meant that if a student missed one step, it would be very difficult to follow the rest of the explanation. The teacher also focused mostly on telling students what would happen under certain conditions, without spending much time explaining why it might happen or giving examples to develop the idea further. Since the student had already made a mistake on the problem, I felt that the part he needed most was probably exactly the reasoning that had been skipped.

These two experiences have strongly influenced the way I think about mathematics teaching. For me, a clear explanation is not simply giving students the correct procedure. Students need enough background knowledge, clear mathematical language and notation, and an explanation of why the mathematics works. They also need connections and examples that help them build a complete picture of the idea.